Mathematical Shell

7/28/2019 Mathematical Shell

1/39

L10/1

LECTURE 10/11: THIN SHELL

1 Equilibrium equations of thin shell

2 Constitutive equations of thin shell

3 Shell examples


2/39

L10/2

SHELL

Para

A thin body in one direction. Curved version of the plate model.

www.modot.org/newsroom/images/Planetarium.J PG www.scottspeck.com/.../north_point/DSCN3526a.jpg


3/39

L10/3

LEARNING OUTCOMES

Student knows the kinematic and kinetic assumptions of the shell model and the

mathematical tools and concepts needed when the solution domain is a surface of

Euclidean three-space, and

is able to derive the component forms of the equilibrium equations from the invariant

vector forms using directed derivatives and Christoffel symbols of midsurfacegeometry, and

is able to derive the component forms of constitutive equations when shell is assumedto be very thin.


4/39

L10/4

THIN BODY DISPLACEMENT ASSUMPTIONS

Bar: 0( , , ) ( )u x y z u x

String: 0( , , ) ( )u s n b u s

Straight beam: 0( , , ) ( ) ( ) ( , )u x y z u x x y z

Curved beam: 0( , , ) ( ) ( ) ( , )u s n b u s s n b

Thin slab: 0( , , ) ( , )u x y z u x y

Membrane: 0( , , ) ( , )u z s n u z s

Plate: 0( , , ) ( , ) ( , ) ( )u x y z u x y x y z

Shell: 0( , , ) ( , ) ( , ) ( )u n u n


5/39

L10/5

SHEL L MODEL ASSUMPTIONS

K inematic assumption: Straight line segments perpendicular to the mid-surface remain

straight in deformation or straight and perpendicular to the mid-surface in deformation.

Therefore displacement ( , , ) ( ) ( ) ( )n n nu n u e u e u e e e ne

.

K inetic assumption: Stress component 0nn .

fdV

tdA

tdA

thin curved body


6/39

L10/6

SHELL EQUILIBRIUM EQUATIONS

( : )( ) 0nF I e F b

in

[ ( : )( ) ] 0n n nM I e M e F c e

in

cS S

in

0n F F

or 0 0 0u u

on

( ) 0nn M M e

or 0

on

in which c( )F J D dn

&

c( )M nJ D dn

&

( )S J dn

Gradient scaling

Volume element scaling

curvature effect

e

e

ne


7/39

L10/7

SHELL EQUATIONS IN CYLINDRICAL GEOMETRY

, ,1 0z zz z zN N bR

in , ,1 0z zM M Q cR

in

, ,1 1 0z zN N Q bR R in , ,1 0zz z z z zM M Q cR

in

, ,1 1 0z z nQ Q N b

R R in

0n F F

or 0 0 0u u

on

( ) 0nn M M e

or 0

on

Boundary conditions are simplest in the rotated system (belonging to ).

ze

e

ne


8/39

L10/8

In the cylindrical geometry and z n coordinates, the directed derivative, the non-zero

Christoffel symbol and curvature components are

z zd

& 1dR

& 1n n R

& 1R

The component forms of equation ( : )( ) 0nF I e F b

are

, ,1 1 1 0i iz kik iz ikz ik kk n z z zz z nz nz zd F F F F b F F F F bR R R

, ,1 1 0i i kik i ik ik kk nz z z nd F F F F b F F F bR R

, ,1 1 0i in kik in ikn ik kk nn n n zn z nd F F F F b F F F bR R


9/39

L10/9

The component forms of equation [ ( : )( ) ] 0n n nM I e M e F c e

( ) 0i ij kik ij ikj ik kk nj nj j jnzd M M M M F c e

, ,

1 1 1 1 0z z n n n nM M M M M F cR R R R

( ) 0i ij kik ij ikj ik kk nj nj j jnd M M M M F c e

, ,1 1 1 0zz z z nz nz nz zM M M M F cR R R

Term nM will be dropped as constitutive equation for a linearly elastic material give

0nM . The component forms of algebraic equations cS S

will be satisfied a

priori by the selection of constitutive equation.


10/39

L10/10

CONSTITUTIVE EQUATIONS IN CYLINDRICAL GEOMETRY

If shell is assumed thin in the sense that only the leading term in t is retained and the

origin of the n axis is placed at the midsurface, constitutive equations simplify to

,

, ,

2

,

,

,

( ))

1(1 )( ) / 2

(z z n

z z nzz

z z z

uu u uN Et

N

N u u

u u

& ,

,( )z n z z

n

Q utG

Q u u

,

3 ,2

, ,

,2

,

,

,2

,

,

,

,

(1 )( ) / 212(1 )

(1 )( ) /

( )

2

z zzz

z

z

z z

z z

n

z zz z

z

z

uMM t EM

M

u u

u

u

&z

z

The blue (quite important) terms are omitted in p.430 of J .N Reddy.

Kirchhoff constraints


11/39

L10/11

EXAMPLE L10/P1. Consider a cylindrical shell of radius R subjected to normal force

nb . Assuming that the free ends are clamped, lengthening in the z direction is not

constrained, and that the solution is rotation symmetric (derivatives with respect to

vanish and 0u ), derive the differential equation and the boundary conditions for

deflection ( ) ( )nw z u z . Material is linearly elastic with properties E and .

Answer:

3

, 222 , 0( )12(1 )zz nzzz z

t E w bR

Etw wR

]0, [z L

, 0zw w

{0, }z L

z

t

x

y

L

R


12/39

L10/12

As derivatives with respect to the angular coordinate vanish, the equilibrium equations

simplify to (after elimination of the shear forces)

, 0zz zN & , ,1 0z z z zN R

M & ,1 0zz zz nN bR

M

The Kirchhoff constraints simplify to 0 and ,z n zu . When rotation variables

are eliminated there, the constitutive equations for the non-zero stress components

simplify to

,21( )

1zz z z n

EtN u u

R

& 2 ,

1( )1

z z nu uEt

NR

3

,2 ,( )12(1 )1

zz n zz z zt

uE

M uR

&3

,2 2( )12(1 )1

n z nzt E

M u uR


13/39

L10/13

The solution is obtained as follows ( , 0zz zN and 0zzN at the free end of the

cylinder and therefore 0zzN )

,21( ) 0

1zz z z n

EtN u u

R

,

1z z nu u

R

2 22

,1 1 1( ) ( )

1 1z z n nn n

Et Etu u u

EtN u u

R R R R

3 3

, ,2 2, 2( ) ( )12(1 ) 12(11

)z zzz n zz nn zz

t E t EM u u u

R Ru

The last equilibrium equation gives

,1 0zz zz nN bR

M

3

, 222 , 0( )12(1 )zz nzzz z

t E w bR

Etw wR


14/39

L10/14

CURVATURE

Curvature is the amount by which a geometric object deviates from being flat, or straight

in the case of a line. Curvature of a surface ( 1/ R ) at a point depends on the direction

of a curve through the point.

Curvature: c( )ne

Principal curvatures: 1 1( , )n and 2 2( , )n such that n n

Gaussian curvature: 1 2det[ ]K

Mean curvature: 1 21 1 1: ( )2 2 2n

H e I

Curvature concept has many somewhat different aspects and the related definitions!

This will be needed later!


15/39

L10/15

K ELVIN-STOKES THEOREM

Kelvin-Stokes theorem relates the surface integral of the curl of a vector field over a

surface S in Euclidean three-space to the line integral of the vector field over its boundary

S . Gauss theorem is obtained as a particular case.

( ) nS Sa e dS a dr

[( ) ( )( )] ( )n nS Sb e b e dS n b ds

The second term on the left hand side takes into account the curvature of the midsurface.

The term vanishes if 0ne or 0nb e (in surface vector).

ne

n dr

S

Scurvature!


16/39


17/39

L10/17

Selection c a b

, vector identity c( ) ( ) : ( )a b a b a b

and definition of

curvature :ne I give a useful integral identity

[( ) ( )( )] ( )n nS Sc e c e dS n c ds

[( ( ) ( )( ) ] ( )n nS SF u e e F u dS n F u ds

c( ) : ( ) ( )( ) ( )n nS S S S

F udS F u dS e e F udS n F u ds

c: ( ) [ ( : )( )] ( )nS S SF u dS F I e F udS n F u ds

The last form can be taken as integration by parts formula on a curved surface. If : 0I (twice the mean curvature) or 0ne F , the usual form used already in

connection with plates is obtained.


18/39

L10/18

VOLUME AND SURFACE ELEMENTS OF SHEL L

In virtual work expression, volume and surface elements of the body dV and dA need to

be expressed in terms of the midsurface area and boundary elements d , d and

thickness element dn . The expressions depend on scaling factors according to:

( )dV J n dnd ( )ndA J n d dA J dnd dA J dnd

Cylinder (1 )n

JR

(1 )nn

JR

(1 )zn

JR

1 J

Sphere 2(1 )n

J R 2(1 )nn

J R (1 )n

J R (1 )n

J R

On a flat surface, curvature vanishes and all the scaling factors have the value 1!


19/39

L10/19

GRADIENT AND MIDSURFACE GRADIENT

In virtual work expression, gradient concerning a generic material point needs to be

expressed in terms of the midsurface gradient and scaling dyad D according to

( ) ( )1

s s s s n s s bs n b

b

bn

e e e e ne ee e e D

s n b

1 1( ) ( )1z z n n z ne e e e e e e e e Dn z R n

1 1 1 1( ) ( )sin1 1 n n n

e e en n

e e e e e e DR R n

Scaling dyad D plays an important role in the constitutive equation. In the very thin body

limit D I

.

midsurface gradient


20/39

L10/20

VIRTUAL WORK DENSITY OF SHELL

int0 c c: ( ) : ( ) ( ) :nw F u M e F S

ext0w b u c

ext0w F u M

in which

c( )F J D dn

&

c( )M nJ D dn

&

S J dn

In contrast to the membrane and plate settings, the number of stress resultants is 3. When a

shell is thin in the sense / 1t R D I

and S F .

e

e

ne


21/39

L10/21

The basic kinematical quanties need to be expressed in terms of the kinematical

quantities of the midsurface

: D

dV J dnd & n ndA d

& dA ndd & dA ndd

Strain ( 0u u n

and displacement gradient is divided into symmetric and

antisymmetric parts with c ) takes the form

0( )nu D u n e

With vector identities :( ) ( ):a b c a b c

, : ( ) ( ):a b c a b c

and c c: :a b a b

the virtual work density of internal forces becomes

midsurface gradient


22/39

L10/22

intc 0 c:[ ( )] :V nw D u n e

intc 0 c c: ( ) : ( ) : ( ) :V nw D u nD D e

intc 0 c c( ): ( ): ( ): :V nw D u n D D e

intc 0 c c c c( ): ( ) ( ): ( ) ( ) ( ):V nw D u nD e D

Grouping all quantities depending on n , writing the volume element in the form

dV J dnd in which d is the midsurface area element, and integration over the

small dimension (thickness) gives

intc 0 c c c[ ( ): ( ) ( ): ( )W J D dn u nJ D dn


23/39

L10/23

c( ) ( ): ]ne J D dn J dn d

int0 c c[ : ( ) : ( ) : ]nW F u M e F S d

c( )F J D dn

& c( )M nJ D dn

& ( )S J dn

Later, the constitutive equations following from definitions of stress resultants do not

assume symmetry of stress which is taken just as the local form of the momentequilibrium of the 3D elasticity.

Volume and area forces contribute to the virtual work of external forces (we omit the

distributed moments although they make sense in the formulation). The surface

contribution needs to be divided into parts coming from the outer and inner surfaces

and from the edge


24/39

L10/24

ext

0[ ( ) ]

VW f udV f u nf J dn d

ext0[ ( ) ]nA nAW t udA t u nt J d

ext0w b u c

& ( )nb fJ dn tJ

& ( )nc nfJ dn ntJ

ext [ ( ) ]i iA iAW t udA t u nt J dn d

{ , }i

ext0w F u M

&

F J tdn

&

M nJ tdn

inner and outer surfaces

edges

volume


25/39

L10/25

SHELL EQUILIBRIUM EQUATIONS

( : )( ) 0nF I e F b

in

[ ( : )( ) ] 0n n nM I e M e F c e

in

cS S

in

0n F F

or 0 0 0u u

on

( ) 0nn M M e

or 0

on

in which c( )F J D dn

&

c( )M nJ D dn

&

( )S J dn

Gradient scaling

Volume element scaling

curvature effect

e

e

ne n t


26/39

L10/26

Integration over the solution domain 3 (surface in Euclidean three-space) and its

boundary gives the virtual work expression

0 c c[ : ( ) : ( ) ( ) : ]nW F u M e F S d

0 0( ) ( )b u c d F u M d

Integration by parts in the first two terms with the Stokes theorem (midsurface is not

flat and therefore the simple Gauss theorem is replaced by the Stokes theorem)

c: ( ) [ ( : )( )] ( )nS S S

F u dS F I e F udS n F u ds

c: ( ) [ ( : )( )] ( )nS S SM dS M I e M dS n M ds


27/39

L10/27

on the mid-surface domain gives an equivalent but more usefull form

0[( ( : )( ) ] ] ( : )nW F I e F b u d S d

[ ( : )( ) ( ) ]n nM I e M e F c d

0( ) ( )n F F u d n M M d

If definition ne and the vector identity ( ) ( )a b c a b c

are used there(to recover the original rotation variable), the virtual work expression becomes

0[( ( : )( ) ] ] ( : )nW F I e F b u d S d

{[ ( : )( ) ( ) ] }n n nM I e M e F c e d


28/39

L10/28

0( ) [( ) ]nn F F u d n M M e d

Finally, the principle of virtual work and the basic lemma of variational calculus give

( : )( ) 0nF I e F b

in

[ ( : )( ) ] 0n n nM I e M e F c e

in

cS S

in

and

0n F F

or 0 0 0u u

on

( ) 0nn M M e

or 0

on

replaces the symmetry of stress in 3D elasti

boundary conditions

equilibrium


29/39

L10/29

PLATE CONSTITUTIVE EQUATIONS

Constitutive equations 0( , )F F u and 0( , )M M u

follow e.g. from the generalized

Hookes law, the definition of small strain, and the kinetic and kinematic assumptions of

the model:

02

1:

zF u kEdz

z zM

& k

[ ( )( ) ( )]1 1

EE I kk I kk I kkkk

& c

1 ( )2

E E E

Derivation of the problem dependent part for the laminated plates, orthotropic material,

heterogeneous material, etc. is straightforward, but the expression may be complicated.

problem dependent part last indexpair conjugate


30/39

L10/30

SHELL CONSTITUTIVE EQUATI ONS

Constitutive equations 0( , )F F u

, 0( , )M M u

follow e.g. from the generalized

Hookes law, the definition of small strain, and the kinetic and kinematic assumptions of

the model:

0c2

1( ): n

n u eFD E DJ dn

M n n

& ne

[ ( )( ) ( )]1 1 n n n n n n n n

EE I e e I e e I e e e e

& c

1 ( )2

E E E

in which the integral expression depends on the material properties (elasticity dyad E

should be modified to include 0nn ) positioning of the midsurface (actually the

reference surface), thickness of the shell, and curvature of the reference surface.

problem dependent part last indexpairconjugate


31/39

L10/31

The small strain expression of the plate making use of the division of the displacement

gradient into symmetric and anti-symmetric part according to u was earlierfound to be (note that c

)

0 nu u e n

.

As rigid body motion needs to produce zero strain a priori, is eliminated to get the

strain expression

c c 0 0 c c1 1 1 1 1: ( ) [ ( ) ] [ ( ) ] ( ) [ ( ) ]2 2 2 2 2n n

u u u u e e n

The stress-strain relationship is taken to consist of a symmetric part depending on

strain and on an anti-symmetric part to be chosen so that moment equilibrium cS S

is


32/39

L10/32

satisfied (the anti-symmetric part is a kind of constraint stress). Assuming linearly

elastic material and using identity : ( ) ( ):a b c a b c

c c1 1: : [ ( ) ] ( ) : :2 2

E E u u E E u E u

: ( ) ( ) :E D u E D u

0: ( ) ( ) : ( )nE D u E D u e n

where c( ) / 2E E E

and cE E

. Notations cE

and

cE denote the first and last

index pair conjugates of E . The precise definitions arec c

: :b a b a

and

c c: :a b a b

b .

The stress-resultant definitions of the virtual work expression give


33/39

L10/33

c c cc 02

c c c c :nD E D nD E D D IF D u e

M nD J dn nD E D n D E D nD I J dnS E D nE D I

in which the integral term depends on the material properties, position of the

midsurface (actually the reference surface), thickness of the shell, and curvature of the

reference surface.

The symmetry condition c 0S S

can be manipulated into the form

0c c( ) 1 : 2 0nu eS S E E D n J dn J dn


34/39

L10/34

the term c c c cc c c2( )E E E E E E E E

vanishes at least when the material

is isotropic as then c c c c2 ( ) 0E E G I I

. Then 0 and the constitutiveequation simplifies to

0cc2

c1( ): nn u eDF J dn D E DJ dn

nDM n n

Assuming a very thin shell so that D I

and 1 J , homogeneous material, and thatthe first moment of n vanishes, the shell expressions boil down to the same form as the

plate expression. Without simplifications the membrane and bending modes are

connected.


35/39

L10/35

CONSTITUTIVE EQUATIONS IN CYLINDRICAL GEOMETRY

If shell is assumed thin in the sense that only the leading term in t is retained and the

origin of the n axis is placed at the midsurface, constitutive equations simplify to

,

, ,

2

,

,

,

( ))

1(1 )( ) / 2

(z z n

z z nzz

z z z

uu u uN

EtN

N u u

u u

& ,

,( )z n z z

n

Q utG

Q u u

,

3 ,2

, ,

,2

,

,

,2

,

,

,

,(1 )( ) / 212(1 )(1 )( ) /

( )

2

z zzz

z

z

z z

z z

n

z zz z

z

z

uMM t EM

M

u u

uu

& z

z

The blue (quite important) terms are omitted in p.430 of J .N Reddy.


36/39

L10/36

Derivation of the stress-strain relationship is a straightforward but somewhat tedious

task. If the Taylor series with respect to thickness are truncated after the second orderterms (third order terms are missing so that remainders 4( )t )

2, , ,

1/ ( ) 12zz z z n z zF E u u u t

2 2, , , ,

1/ ( ( )12

)z z n nF E u u u t u u

2, , ,1/ ( ) 12z z z zF Gt u u t

2 2, , , ,

1/ ( ) (1

)2z z z z z

F Gt u u t u

,/ ( ) / ( )zn nz z n zF Gt F Gt u

, , ,/zz z z z zM D u


37/39


38/39

L10/38

CONSTITUTIVE EQUATI ONS IN SPHERICAL GEOMETRY

If shell is assumed thin in the sense that only the leading term in t needs to be retained and

the origin of the n axis is placed at the midsurface, constitutive equations simplify to

, ,

, ,

,

2

,

( cot csc ( )( cot csc ( )

(1 )( cot csc ) / 2

))

1

n n

n n

u u u u uu

Nt E

NR

u u u u

u uN u

3

, ,

, ,

2

, ,cot csc

( cot csc )

(1 )( cot c12 1

sc ) / 2

Mt EM

RM

,

,

( csc )

( )n

n

u uQtG

Q u u

&

& 1csc

sin


39/39

L10/39

EXAMPLE L10/P2. Show that small rigid body motion does not introduce stress in the

constitutive equations of thin cylindrical shell. The displacement and rotation fields of rigid body motion are given by 0 0( , , ) ( )u z n u r

in which 0u and are

constant vectors so that their components are constant in a Cartesian coordinate system.

The components in the cylindrical z n coordinate system are

T

0

( sin cos )

( cos sin ( cos sin ) )cos sin ( sin cos )

x yz z

y x z x yn x y x y

Re u

u e u u R ze u u z

,

T cos siny xzn

z

ee e

. Answer: Stress vanishes.

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